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Title: Reordering for Series
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Series: Real Analysis
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Chapter: Infinite Series
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YouTube-Title: Real Analysis 21 | Reordering for Series
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Bright video: https://youtu.be/GADre0hHc4c
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Dark video: https://youtu.be/M3NFkozMOqo
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra21_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps
00:00 Intro
00:17 Meaning of reordering of series
01:13 Example: Reordering can change the value
02:32 2nd Example with a convergent series
04:58 Definition Reordering
06:10 Theorem for abs. convergent series
06:39 Proof of the Theorem
12:22 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Is the series $\displaystyle \sum_{k = 1}^\infty (-1)^{k+1} \frac{1}{k}$ absolutely convergent?
A1: Yes, it is.
A2: No, it isn’t.
Q2: When do we call a series $\displaystyle \sum_{k = 1}^\infty a_{\tau(k)}$ a reordering of the series $\displaystyle \sum_{k = 1}^\infty a_k$?
A1: It’s called a reordering if $\tau : \mathbb{N} \rightarrow \mathbb{N}$ is map.
A2: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a injective map.
A3: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a surjective map.
A4: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a bijective map.
A5: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a bijective map with $\tau(1) = 1$.
Q3: Can one change the value of a convergent series with a reordering.
A1: Yes, that is always possible.
A2: No, this is never possible.
A3: Yes, this is possible but only if the series is not absolutely convergent.
Q4: Let $q \in \mathbb{R}$ with $|q| < 1$. Can one change the limit of the series $\displaystyle \sum_{k = 0}^\infty q^k$ with a reordering?
A1: Yes, this is always possible.
A2: No, this is not possible.
A3: It possible for $q < 0$ but not for $q \geq 0$.
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Last update: 2025-01
